Liquid Bonds - msolids/musen GitHub Wiki

Bond properties

L_b - length of liquid bridge, is the distance between particle surfaces

R_b – radius of liquid bridge. It is assumed that radius does not change during simulation. Bond radius cannot be larger than particle radius ( $R_b\leq R_1\,; R_b \leq R_2$ )

h_1,h_2 – distances between end point of spheres and wetting points (boundary of wetted/dry surfaces). $h_1\leq R_1\,; h_2\leq R_2$

V_b – volume of liquid bridge. Volume calculated according to the position of particles in the initial time point as:

$h_1=R_1-\sqrt{(R_1^2-R_b^2 )}$

$h_2=R_2-\sqrt{(R_2^2-R_b^2 )}$

$V_b=\pi\cdot R_b^2\cdot (L_b+h_1+h_2)-\frac{\pi\cdot h_1^2 (3R_1-h_1)}{3}-\frac{\pi\cdot h_2^2 (3R_2-h_2)}{3}$

Main equations

$R^*=\frac{R_1\cdot R_2}{R_1+R_2}$

$\bar{v}_{rel}=\bar{v}_2-\bar{v}_1-\frac{(\bar{\omega}_1+\bar{\omega}_2)\times\bar{r}_c}{2}$

$F_{cap}=\exp(A\cdot L_b+B)+C$

$A=-1.1\cdot V_b^{0.53}$

$B=(-0.34\cdot \ln(V_b)-0.96)\cdot\theta-0.019 \ln(V_b)+0.48$

$C=0.0042\cdot \ln(V_b)+0.078$

$F_{vis,n}=v_{rel,n}\cdot 6 \cdot \pi \cdot \mu \cdot \frac{(R^*)^2}{L_b}$ (Zhu et al., 2011; Lian et al., 1998)

$F_{vis,t}=v_{rel,t}\cdot 6 \cdot \pi \cdot \mu \cdot R^* \cdot[\frac{8}{15}\cdot \ln(\frac{R^*}{L_b})+0.9588]$ (Lian et al., 1998)

Force and moment in normal direction

$F_n=F_{cap}+F_{vis,n}$

Force and moment in shear direction

$F_t = F_{vis,t}$

Moments

$M_{tot}=\frac{r_c}{2}\cdot F_{t,b}$

Breakage criteria

Elongation of the bond is used as rupture criteria (Lian et al., 1993):

$L_b>(1+0.5\theta)\sqrt[3]{V_b}$

Minimal bond thickness

In order to avoid very large viscous forces, the bond length is limited by minimal thickness which can be specified by user in model settings ( $L_b>L_{min}$ ).

Literature

Lian G., Thornton C., Adams M. J. (1993). A theoretical study of the liquid bridge forces between two rigid spherical bodies. Journal of Colloidal and Interface Science, 161, 138–147.

Lian G., Thornton C., Adams M. J. (1998). Discrete particle simulation of agglomerate impact coalescence. Chemical Engineering Science 53, 3381–3391.

Mikami T., Kamiya H., Horio M. (1998). Numerical simulation of cohesive powder behavior in a fluidized bed. Chemical Engineering Science, 53, 1927–1940.

Zhu R.R., Zhu W.B., Xing L.C., Sun Q.Q. (2011). DEM simulation on particle mixing in dry and wet particle spouted bed. Powder Technology, 210, 73–81.

Symbol	Description
	Force in normal and tangential directions [N]
$F_{cap}, F_{vis}$	Capillary and viscous force [N]
	Length of liquid bridge [N]
	Particle masses [kg]
$\bar{v}_{rel}$	Relative velocity [m/s]
$\bar{v}_1, \bar{v}_2$	Translational velocities of contact partners [m/s]
	Particle radii [m]
	Equivalent radius [m]
$\bar{r}_c$	Contact vector [m]
$\bar{r}_n$	Normalized contact vector [-]
$\bar{\omega}_1, \bar{\omega}_2$	Rotation velocities of particles [rad/s]
$\mu$	Dynamic viscosity [Pa*s]
$\theta$	Contact angle [rad]